Modeling Tools for Environmental Engineers and Scientists

Figure 3.5 Using MATLAB ® for solving simultaneous equations.advantage of the fact that the function should change sign within the intervalbounded by x I and x u . Or, in other words, guess x I and x u so that:f (x I ,) • f (x u ,) < 0 (3.6)This can be readily achieved by plotting the function. Then, a refined valueof the root, x r , can be estimated as = ( x I + x u )/2. To make the next refinedguess, a new bracket is now defined with either x I and x r or x r and x u . Again,a sign change of f (x) is used to decide which range to make the new guessfrom:if f (x I ,) × f (x r ,) < 0, the new guess is made between x I and x rif f (x r ,) × f (x u ,) < 0, the new guess is made between x r and x u(3.7a)(3.7b)This process is iterated until the new guess is not significantly different fromthe previous one; at that point, the root is taken as the value of the last guess.Worked Example 3.2First-order processes occurring in many environmental systems can bedescribed by the equation: C = C 0 e –kt , where C is the concentration of thechemical undergoing the reaction, C 0 is its initial concentration, k is the reactionrate constant, and t is the time. Find the time it would take for the concentrationto drop from 100 mg/L to 10 mg/L.SolutionEven though the equation can be solved for t algebraically, the binarymethod is used here to illustrate the method and to compare its performanceagainst the direct algebraic solution. The algebraic solution can be readilyseen as t = 9.21. The implementation of the binary algorithm in an Excel ®spreadsheet is shown in Figure 3.6.© 2002 by CRC Press LLC